Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A variational method for functions of bounded boundary rotation


Author: H. B. Coonce
Journal: Trans. Amer. Math. Soc. 157 (1971), 39-51
MSC: Primary 30.43
DOI: https://doi.org/10.1090/S0002-9947-1971-0274737-8
MathSciNet review: 0274737
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ f$ be a function analytic in the unit disc, properly normalized, with bounded boundary rotation. There exists a Stieltjes integral representation for $ 1 + zf''(z)/f'(z)$. From this representation, and in view of a known variational formula for functions of positive real part, a variational formula is derived for functions of the form $ q(z) = 1 + zf''(z)/f'(z)$. This formula is for functions of arbitrary boundary rotation and does not assume the functions to be univalent.

A new proof for the radius of convexity for functions of bounded boundary rotation is given. The extremal function for Re$ \{ F(f'(z))\} $ is derived. Examples of univalent functions with arbitrary boundary rotation are given and estimates for the radius in which Re$ \{ f'(z)\} > 0$ are computed.

The coefficient problem is solved for $ {a_4}$ for all values of the boundary rotation and without the assumption of univalency.


References [Enhancements On Off] (What's this?)

  • [1] D. A. Brannan, On functions of bounded boundary rotation. II, Bull. London Math. Soc. 1 (1969), 321-322. MR 0264046 (41:8643)
  • [2] J. A. Hummel, A variational method for starlike functions, Proc. Amer. Math. Soc. 9 (1958), 82-87. MR 20 #1779. MR 0095273 (20:1779)
  • [3] W. E. Kirwan, A note on extremal problems for certain classes of analytic functions, Proc. Amer. Math. Soc. 17 (1966), 1028-1030. MR 34 #2854. MR 0202995 (34:2854)
  • [4] O. Lehto, On the distortion of conformal mappings with bounded boundary rotation, Ann. Acad. Sci. Fenn. Ser. A. I. Math. Phys. No. 124 (1952). MR 14, 743. MR 0053241 (14:743a)
  • [5] V. Paatero, Über die konforme Abbildung von Gebieten deren Ränder von beschränkter Drehung sind, Ann. Acad. Sci. Fenn. Ser. A33 No. 9 (1931).
  • [6] -, Über Gebiete von beschränkter Randdrehung, Ann. Acad. Sci. Fenn. Ser. A37 No. 9 (1933).
  • [7] B. Pinchuk, Extremal problems for functions of bounded boundary rotation, Bull. Amer. Math. Soc. 73 (1967), 708-711. MR 35 #6827. MR 0215992 (35:6827)
  • [8] M. S. Robertson, Variational methods for functions with positive real part, Trans. Amer. Math. Soc. 102 (1962), 82-93. MR 24 #A3288. MR 0133454 (24:A3288)
  • [9] -, Extremal problems for analytic functions with positive real part and applications, Trans. Amer. Math. Soc. 106 (1963), 236-253. MR 26 #325. MR 0142756 (26:325)
  • [10] -, Coefficients of functions with bounded boundary rotation, Canad. J. Math. 21 (1969), 1477-1482. MR 0255798 (41:458)
  • [11] -, The sum of univalent functions, Duke Math. J. 38 (1970), 411-419. MR 0264052 (41:8649)
  • [12] K. Sakaguchi, A variational method for functions with positive real part, J. Math. Soc. Japan 16 (1964), 287-297. MR 31 #1375. MR 0177111 (31:1375)
  • [13] M. Schiffer and O. Tammi, A method of variations for functions with bounded boundary rotation, J. Analyse Math. 17 (1966), 109-144. MR 35 #5601. MR 0214752 (35:5601)
  • [14] -, On the fourth coefficient of univalent functions with bounded boundary rotation, Ann. Acad. Sci. Fenn. Ser. A. I. No. 396 (1967). MR 35 #3052. MR 0212177 (35:3052)
  • [15] O. Tammi, Note on symmetric schlicht domains of bounded boundary rotation, Ann. Acad. Sci. Fenn. Ser. A. I. No. 198 (1955). MR 17, 599. MR 0074516 (17:599e)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 30.43

Retrieve articles in all journals with MSC: 30.43


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0274737-8
Keywords: Bounded boundary rotation, coefficient problems, extremal problems, radius of convexity, univalent functions, variational methods
Article copyright: © Copyright 1971 American Mathematical Society

American Mathematical Society